Question

Find measures of all the angles.Measures for angles 1 to 32

Answer 1

According to the given graph, we have the following.

Angle 7 is 24°, given.

Angle 20 is 80°, given.

Angle 7 and Angle 11 sum 180°, by same-side interior angles theorem.

`\begin{gathered} m\angle7+m\angle11=180 \\ 24+m\angle11=180 \\ m\angle11=180-24=156 \end{gathered}`

Angle 11 measures 156°.

Angle 11 and angle 14 are equal, by vertical angles theorem.

`m\angle14=m\angle11=156`

Angle 14 measures 156°.

Angle 12 and angle 14 sum 180°, by supplementary angles theorem.

`\begin{gathered} m\angle12+m\angle14=180 \\ m\angle12+156=180 \\ m\angle12=180-156=24 \end{gathered}`

Angle 12 measures 24°.

Angle 13 and angle 12 are equal, by vertical angles theorem.

`m\angle13=m\angle12=24`

Angle 13 measures 24°.

Angle 24 and angle 12 are equal, by corresponding angles.

`m\angle24=m\angle12=24`

Angle 24 measures 24°.

Angle 26 is equal to angle 14, by corresponding angles.

`m\angle26=m\angle14=156`

Angle 26 measures 156°.

Angle 23 is equal to angle 11, by corresponding angles.

Angle 23 measures 156°.

Angle 25 is equal to angle 13, by corresponding angles.

Angle 25 measures 24°.

Angle 29 is equal to angle 25, by alternate interior angles.

Angle 29 measures 24°.

Angle 30 is equal to angle 25, by corresponding angles.

Angle 30 measures 24°.

Angle 8 is equal to angle 30, by corresponding angles.

Angle 8 measures 24°.

Angle 22 is equal to angle 20, by vertical angles.

Angle 22 measures 80°.

Angles 20 and 19 sum 180°, by supplementary angles.

`\begin{gathered} m\angle19+m\angle20=180 \\ m\angle19+80=180 \\ m\angle19=180-80=100 \end{gathered}`

Angle 19 measures 100°.

Angle 21 is equal to angle 19, by vertical angles.

Angle 21 measures 100°.

Angle 5 and angle 19 are equal, by corresponding angles.

Angle 5 measures 100°.

Angle 10 and angle 5 are equal, by vertical angles.

Angle 10 measures 100°.

Angles 8, 9, and 10 sum 180°, by supplementary angles.

`\begin{gathered} m\angle8+m\angle9+m\angle10=180 \\ 24+m\angle9+100=180 \\ m\angle9=180-100-24=56 \end{gathered}`

Angle 9 measures 56°.

Angle 6 is equal to angle 9, by vertical angles.

Angle 6 measures 56°.

Angles 10, 13, and 16 sum 180°, by triangle interior angles theorem.

`\begin{gathered} m\angle10+m\angle13+m\angle16=180 \\ 100+24+m\angle16=180 \\ m\angle16=180-24-100=56 \end{gathered}`

Angle 16 measures 56°.

Angle 18 is equal to angle 16, by vertical angles.

Angle 18 measures 56°.

Angle 16 and angle 17 sum 180°, by supplementary angles.

`\begin{gathered} m\angle16+m\angle17=180 \\ 56+m\angle17=180 \\ m\angle17=180-56 \\ m\angle17=124 \end{gathered}`

Angle 17 measures 124°.

Angle 15 and angle 17 are equal, by vertical angles.

Angle 15 measures 124°.

Angle 2 is equal to the sum of angles 6 and 7, by corresponding angles.

`m\angle2=m\angle6+m\angle7=56+24=80`

Angle 2 measures 80°.

Angle 3 is equal to angle 2, by vertical angles.

Angle 3 measures 80°.

Angle 1 and angle 2 sum 180°, by supplementary angles.

`\begin{gathered} m\angle1+m\angle2=180 \\ m\angle1+80=180 \\ m\angle1=180-80=100 \end{gathered}`

Angle 1 measures 100°.

Angle 4 is equal to angle 1, by vertical angles.

Angle 4 measures 100°.

Angle 3 is equal to the sum of angles 30 and 31, by corresponding angles.

`\begin{gathered} m\angle3=m\angle30+m\angle31 \\ 80=24+m\angle31 \\ m\angle31=80-24=56 \end{gathered}`

Angle 31 measures 56°.

Angle 28 is equal to angle 31, by vertical angles.

Angle 28 measures 56°.

Angles 30, 27, and 28 sum 180, by supplementary angles.

`\begin{gathered} m\angle30+m\angle27+m\angle28=180 \\ 24+m\angle27+56=180 \\ m\angle27=180-56-24=100 \end{gathered}`

Angle 27 measures 100°.

Angle 32 is equal to angle 27, by vertical angles.

Angle 32 measures 100°.